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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj873 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 31625. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj873.4 | ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} |
bnj873.7 | ⊢ (𝜑′ ↔ [𝑔 / 𝑓]𝜑) |
bnj873.8 | ⊢ (𝜓′ ↔ [𝑔 / 𝑓]𝜓) |
Ref | Expression |
---|---|
bnj873 | ⊢ 𝐵 = {𝑔 ∣ ∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj873.4 | . 2 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} | |
2 | nfv 2015 | . . 3 ⊢ Ⅎ𝑔∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) | |
3 | nfcv 2970 | . . . 4 ⊢ Ⅎ𝑓𝐷 | |
4 | nfv 2015 | . . . . 5 ⊢ Ⅎ𝑓 𝑔 Fn 𝑛 | |
5 | bnj873.7 | . . . . . 6 ⊢ (𝜑′ ↔ [𝑔 / 𝑓]𝜑) | |
6 | nfsbc1v 3683 | . . . . . 6 ⊢ Ⅎ𝑓[𝑔 / 𝑓]𝜑 | |
7 | 5, 6 | nfxfr 1954 | . . . . 5 ⊢ Ⅎ𝑓𝜑′ |
8 | bnj873.8 | . . . . . 6 ⊢ (𝜓′ ↔ [𝑔 / 𝑓]𝜓) | |
9 | nfsbc1v 3683 | . . . . . 6 ⊢ Ⅎ𝑓[𝑔 / 𝑓]𝜓 | |
10 | 8, 9 | nfxfr 1954 | . . . . 5 ⊢ Ⅎ𝑓𝜓′ |
11 | 4, 7, 10 | nf3an 2006 | . . . 4 ⊢ Ⅎ𝑓(𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′) |
12 | 3, 11 | nfrex 3216 | . . 3 ⊢ Ⅎ𝑓∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′) |
13 | fneq1 6213 | . . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑛 ↔ 𝑔 Fn 𝑛)) | |
14 | sbceq1a 3674 | . . . . . 6 ⊢ (𝑓 = 𝑔 → (𝜑 ↔ [𝑔 / 𝑓]𝜑)) | |
15 | 14, 5 | syl6bbr 281 | . . . . 5 ⊢ (𝑓 = 𝑔 → (𝜑 ↔ 𝜑′)) |
16 | sbceq1a 3674 | . . . . . 6 ⊢ (𝑓 = 𝑔 → (𝜓 ↔ [𝑔 / 𝑓]𝜓)) | |
17 | 16, 8 | syl6bbr 281 | . . . . 5 ⊢ (𝑓 = 𝑔 → (𝜓 ↔ 𝜓′)) |
18 | 13, 15, 17 | 3anbi123d 1566 | . . . 4 ⊢ (𝑓 = 𝑔 → ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))) |
19 | 18 | rexbidv 3263 | . . 3 ⊢ (𝑓 = 𝑔 → (∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′))) |
20 | 2, 12, 19 | cbvab 2952 | . 2 ⊢ {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} = {𝑔 ∣ ∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)} |
21 | 1, 20 | eqtri 2850 | 1 ⊢ 𝐵 = {𝑔 ∣ ∃𝑛 ∈ 𝐷 (𝑔 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ w3a 1113 = wceq 1658 {cab 2812 ∃wrex 3119 [wsbc 3663 Fn wfn 6119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4875 df-opab 4937 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-fun 6126 df-fn 6127 |
This theorem is referenced by: bnj849 31542 bnj893 31545 |
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